Argument Principle

If  is meromorphic in a region  enclosed by a contour , let  be the number of complex roots of  in , and  be the number of poles in , with each zero and pole counted as many times as its multiplicity and order, respectively. Then

Defining  and  gives

REFERENCES:

Duren, P.; Hengartner, W.; and Laugessen, R. S. "The Argument Principle for Harmonic Functions." Amer. Math. Monthly 103, 411-415, 1996.

Knopp, K. Theory of Functions, Parts I and II. New York: Dover, pp. 132-134, 1996.

Krantz, S. G. "The Argument Principle." Ch. 5 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 69-78, 1999.